Nuclear Waste Containment Test Problems for Numerical Homogenization Methods
Contents:
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Map 1 Image
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Map 2 Image
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Problem Description
This page describes the nuclear waste containment test problem for testing
homogenization methods. This test problem is based on a real problem in porous
media proposed by several people in France (e.g, Alain Bourgeat). The data files
contain conductivity coefficients defined on a relatively coarse grids. The test
involves the behavior of contaminant transport of the homogenized maps created
by various method. The idea is to homogenize a single containment cell and then
consider a number of these next to each other each containing some amount of
nuclear material.
Problem/Data Links:
The following is a list of raw data sets that can be downloaded and used as
input for your own homogenization codes for testing, input to a heat equation
solver or other applications. These maps can also be downloaded and used as
input for the JHomogenizer application. In this example original .pdf files are
included to show the configuration of the values. These images were used to
build the data sets.
- Data Files:
- Map 1 .pdf file This file will show the
original configuration of a single containment cell.
- Map 2 .pdf file This file will show the
original configuration of a single containment cell.
- pdf Files:
- Map 1 Data This the actual map data file
as depicted in the file above.
- Map 2 Data This the actual map data file
as depicted in the file above.
Homogenization Results:
The following documents the results for various homogenization methods applied
to the maps shown above. These results are shown for homogenization applied to
the entire region shown in the maps. The results for the two maps are shown
separately. One could use these results to simulate flow through a group of
these patterns that might make up a nuclear waste repository. One example of
such a simulation is shown below.
Map 1 Homogenization Results:
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Homogenization Method:
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Homogenization Tensor:
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Arithmetic Average:
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| 0.689891E-07 |
0.0000 |
0.0000 |
| 0.0000 |
0.689891E-07 |
0.0000 |
| 0.0000 |
0.0000 |
0.689891E-07 |
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Geometric Average:
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| 0.275560E-07 |
0.0000 |
0.0000 |
| 0.0000 |
0.275560E-07 |
0.0000 |
| 0.0000 |
0.0000 |
0.275560E-07 |
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Harmonic Average:
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| 0.393582E-08 |
0.0000 |
0.0000 |
| 0.0000 |
0.393582E-08 |
0.0000 |
| 0.0000 |
0.0000 |
0.393582E-08 |
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HomCode Average:
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| 0.371243E-07 |
-0.335548E-14 |
0.0000 |
| -0.335548E-14 |
0.540459E-08 |
0.0000 |
| 0.0000 |
0.0000 |
0.689891E-07 |
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Linear Boundary Condition Average:
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| 6.6682 |
0.0000 |
0.0000 |
| 0.0000 |
6.6682 |
0.0000 |
| 0.0000 |
0.0000 |
7.7500 |
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Fractured Media Tensor Average:
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Not Applicable
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Linear Boundary Condition Wavelet Average:
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Not Applicable
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Periodic Wavelet Average:
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Not Applicable
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Map 2 Homogenization Results:
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Homogenization Method:
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Homogenization Tensor:
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Arithmetic Average:
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| 0.801743E-08 |
0.0000 |
0.0000 |
| 0.0000 |
0.801743E-08 |
0.0000 |
| 0.0000 |
0.0000 |
0.801743E-08 |
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|
Geometric Average:
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| 0.735949E-10 |
0.0000 |
0.0000 |
| 0.0000 |
0.735949E-10 |
0.0000 |
| 0.0000 |
0.0000 |
0.735949E-10 |
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Harmonic Average:
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| 0.151819E-10 |
0.0000 |
0.0000 |
| 0.0000 |
0.151819E-10 |
0.0000 |
| 0.0000 |
0.0000 |
0.151819E-10 |
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HomCode Average:
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| 0.478375E-10 |
-0.560156E-14 |
0.0000 |
| -0.560156E-14 |
0.224193E-10 |
0.0000 |
| 0.0000 |
0.0000 |
0.689891E-07 |
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Linear Boundary Condition Average:
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| 6.6682 |
0.0000 |
0.0000 |
| 0.0000 |
6.6682 |
0.0000 |
| 0.0000 |
0.0000 |
7.7500 |
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Fractured Media Tensor Average:
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Not Applicable
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Linear Boundary Condition Wavelet Average:
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Not Applicable
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Periodic Wavelet Average:
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Not Applicable
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Simulation Results:
The following graphics show results for simulations performed on a heterogeneous
maps and the homogenized maps that result from applying various methods of
averaging. The heterogeneous map used in the generation of the flow simulations
looks like the map above. In this example two of the waste repositories are
placed in a line as might be the case in a real waste storage site. The maps
were generated by the JHomogenizer tool. The graphics below were generated using
the JHomogenizer tool.
Heterogeneous Simulation Results
The results below show the solution of the elliptic equation for the original
heterogeneous map provided in the heating element data set. These are shown to
compare with the results with solutions computed on homogenized versions of the
heating element data.
Heterogeneous Simulation Results
The following simulation shows an elliptic solution for a single cell of the
form shown above. The results are shown for both of the maps.
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Map 1 Elliptic Solution - One Cell
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Map 2 Elliptic Solution - One Cell
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The two cells maps look like the two maps shown below. The results from the
solution of the elliptic problems with the heterogeneous maps are shown below
the maps. Note that the images have been compressed in the vertical direction.
This is done to save a bit of space. The actual size in the vertical direction
is twice the size of the one cell map shown near the top of this web page.
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Map 1 Image - Two Cell
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Map 2 Image - Two Cell
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Map 1 Elliptic Solution - Two Cell
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Map 2 Elliptic Solution - Two Cell
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Homogenized Simulation Results with 2 Cells:
The following results show the elliptic solutions for the homogenized versions
of the 2 cell problem as shown above. The results for Map 1 are presented first
followed by the results for Map 2.
Map 1 Simulation Results with 2 Cells:
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Arithmetic Average
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Geometric Average
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HomCode Average
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Harmonic Average
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Map 2 Simulation Results with 2 Cells:
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Arithmetic Average
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Geometric Average
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HomCode Average
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Harmonic Average
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Steps to use JHomogenizer to create and work with the heating element map:
To create the results included on these pages and to get started working on
other applications you can use the JHomogenizer tool.